Answer: I and III only

## Explanation The F-test in regression analysis is used to test linear restrictions on the coefficients. The key requirement is that the null hypothesis must be expressed as **linear combinations** of the regression coefficients. Let's analyze each hypothesis: **I. β₁ = 1** - This can be written as: β₁ - 1 = 0 - This is a linear restriction (linear in β₁) - ✅ **Can be tested using F-test** **II. β₂² = 1** - This involves a squared term: β₂² - This is **not** a linear restriction - ❌ **Cannot be tested using F-test** **III. β₃ = -2β₂** - This can be written as: β₃ + 2β₂ = 0 - This is a linear combination of β₃ and β₂ - ✅ **Can be tested using F-test** **IV. β₁β₂ = 0** - This involves a product of two coefficients - This is **not** a linear restriction - ❌ **Cannot be tested using F-test** Therefore, only hypotheses I and III can be tested using an F-test. **Key Concept**: The F-test requires that the null hypothesis consists of **linear restrictions** on the regression coefficients. Any hypothesis involving non-linear operations (squares, products, etc.) cannot be tested using the standard F-test.

Author: Nikitesh Somanthe